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ARTICLE IN PRESS

Engineering Structures xx (xxxx) xxx–xxxwww.elsevier.com/locate/engstruct

Development of a closed-form 3-D RBS beam finite element and associatedcase studies

Samuel Kinde Kassegne∗

Department of Mechanical Engineering, College of Engineering, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-1323, United states

Received 15 January 2006; received in revised form 9 September 2006; accepted 11 September 2006

Abstract

The 1994 Northridge and the 1995 Kobe earthquakes demonstrated that steel moment resisting frames (SMRF) which were traditionallyconsidered to provide an ideal structural system in resisting lateral earthquake loads could indeed suffer brittle fractures at the beam–columnconnections. To remedy this, competing connection design improvements have been adopted, among which is the radius-cut Reduced BeamSection (RBS) moment connection. As the design of a vast majority of these SMRF buildings is drift-controlled, the reduction in lateral stiffnessand hence the increase in drifts in general 3-dimensional SMRF buildings with RBS moment connections – particularly those with torsionalirregularity – is investigated here through development and verification of a new closed-form 3-D RBS finite element based on Timoshenko’sshear deformation theory for beams. The study shows that the increase in story drifts in 3-dimensional torsionally irregular SMRF buildings withRBS moment connections could be as high as 15% under the action of lateral and gravity loads. This is in contrast to the current recommendationof increasing elastic story drifts by a maximum of 9%–10% for SMRF buildings to account for stiffness reduction in the presence of RBSconnections. The study also shows that the decrease in weak axis joint rotational and shear stiffness coefficients of beams with RBS connectionson both sides – and unsupported laterally – could be as high as 67% for flange area reductions of only 30%.c© 2006 Published by Elsevier Ltd

Keywords: Earthquake design; RBS sections; Frames; Connections; Drift; Irregular buildings; Stiffness; Earthquakes

1. Introduction1

One of the most important outcomes of the structural2

investigations of damages in the Northridge earthquake of3

1994 was the finding of brittle fracture failures in welds at4

beam–column connections in as many as 200 steel special5

moment resisting frame (SMRF) buildings in the Los Angeles6

area [24]. While no actual collapse of steel moment frame7

buildings occurred, greater damage and even total collapse of8

these buildings could have resulted if the ground-shaking had9

continued for a longer time. The 1995 Kobe earthquake that hit10

the city of Kobe, Japan, a year after the Northridge earthquake11

had much more adverse consequences where a number of steel12

moment frame buildings suffered partial or total collapse, again13

confirming the fact that steel moment connections were not as14

safe as they were previously believed to be.15

∗ Tel.: +1 760 402 7162; fax: +1 619 594 3599.E-mail addresses: kassegne@mail.sdsu.edu, samk@digitaladdis.com.

As a direct consequences of these damages, a number of 16

initiatives were forwarded to investigate the causes for the 17

poor performance of moment connections and to come up 18

with improved seismic design and retrofitting techniques for 19

steel moment frames. Out of these initiatives, a number of 20

competing ‘prequalified’ design improvements in beam column 21

connections in special steel moment resisting frames have 22

found their way to the design practice. One of the design 23

improvements uses what is called Reduced Beam Section 24

(RBS) or dog-bone moment connection and is a prequalified 25

connection [7] in which a portion of the top and bottom flanges 26

is cut out at some distance from the face of columns. The RBS 27

connection is essentially intended to pull the plastic hinge away 28

from the face of the column where the stress and strain demands 29

may cause the weld to fracture. 30

In the past several years, a number of steel buildings 31

with SMRF have increasingly made use of ‘prequalified’ 32

RBS moment connections. A number of structural engineering 33

design firms in the US and Pacific Rim countries now design 34

0141-0296/$ - see front matter c© 2006 Published by Elsevier Ltddoi:10.1016/j.engstruct.2006.09.010

Please cite this article in press as: Kassegne SK. Development of a closed-form 3-D RBS beam finite element and associated case studies. Engineering Structures(2006), doi:10.1016/j.engstruct.2006.09.010

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Notations

b f width of beam flange;t f thickness of beam flange;d depth of beam;L total length of beam;Imaj major moment of inertia;Imin minor moment of inertia;A area of cross section;Asmaj major shear area;Asmin minor shear area;J polar moment of inertia;W virtual work done by a unit load;R radius of flange cut;r ratio of reduced flange width to the original flange

width;a distance from face of column to beginning of

radius cut;b half-width of radius cut;c maximum depth of radius cut;E Young’s modulus;G Shear modulus.

such SMRF frames with RBS moment connections on a routine1

basis. The typical modelling of the effect of RBS moment2

connections on drift for routine analysis and design has been3

based on the ‘10% drift increase’ approach recommended by4

Grubbs [10], Moore et al. [16], and the ‘9% drift increase for5

50% cut’ approach recommended by FEMA [7] and SAC [18].6

The ‘10%’ limit proposed by Moore et al. [16] was based7

on a 2-dimensional analysis results reported by Grubbs [10]8

which do not take into account diaphragm rotations, weak9

axis bending, stiffness irregularities and a combination of10

arbitrary gravity and lateral loads. However, there is no11

adequate information on what study the ‘9%’ limit (for 50%12

radius cut) proposed by the FEMA guideline is based on.13

Further, the SEAOC Seismology Committee commentary and14

recommendations on FEMA 350 is quiet on the subject [19].15

Similarly, there does not seem to be any published reference16

where AISC or any other code bodies question the issue. It17

seems, therefore, that a significant number of moment resisting18

frames with RBS moment connections are currently being19

analyzed based on gross geometry and then story displacements20

adjusted for stiffness reduction by a flat 9% or 10%. In21

a further note, the typical tests on RBS systems such as22

those done by [22,5,4] were based on subassemblages which23

consisted of typically W36 beams and W14 columns and were24

2-dimensional in nature and, consequently, did not shed light25

on elastic diaphragm torsional rotations, weak axis bending of26

beams and the interaction of gravity and lateral loads.27

However, the past few years have witnessed an ongoing28

research to develop rational, accurate and efficient methods29

for the exact analysis of SMRF with RBS connections. Hailu30

et al. [11] had reported a preliminary work that employs the use31

of mixed elements (shell and beam elements) constrained with32

MPC – multi-point constraints – to model the actual geometry33

and stiffness reductions in RBS systems. The method exploits 34

the advantages of solid and shell elements for accuracy and 35

beam elements for economy. Numerical results for realistic 36

building configurations have not been reported yet, however. 37

Further, the computational time and modelling effort required 38

in this approach may turn out to be too prohibitive for 39

routine design purposes. The work of Chambers [2] is the 40

first attempt on closed-form solution for moment frames 41

with RBS connection. Chambers [2] predicted a maximum 42

drift increase of 10.6% for a 2-dimensional frame system 43

with RBS connection of 40% flange reductions. They also 44

reported a maximum decrease of 15.1% in the stiffness 45

coefficient corresponding to joint rotations in frames with RBS 46

connections of 44% reduction in flange area. Their work is, 47

however, limited to two-dimensional frames and is based on 48

classical beam theory which underestimates deflections both in 49

beams and columns. Further, the effect of gravity loads, shear 50

deformations, torsion and minor axis bending were not included 51

in their formulations. 52

To date, therefore, no report exists in the open literature on 53

a complete 3-dimensional frame system analysis – including 54

shear deformation – using exact FEA modelling of RBS 55

members. In the absence of such studies, the effect of 56

3-dimensional loading, shear deformation, weak axis bending 57

and most importantly torsion and a combination of gravity and 58

lateral loads – particularly for moment frames with irregular 59

stiffness distribution – were at best unknown for routine design 60

purposes. 61

In this work, this need is addressed by developing a new 62

and numerically exact stiffness matrix for RBS members 63

based on the potential energy approach for deriving stiffness 64

coefficients. The stiffness coefficients developed here are for 65

a general 3D beam element and include major and minor 66

axis bending, shear deformation and torsion and are based on 67

Timoshenko’s Beam Theory. A simple programme that uses 68

the newly derived RBS stiffness matrix was developed in the 69

course of this research. The only additional input required for 70

each beam with RBS section are the parameters ‘a’, ‘b’ and 71

‘c’ as shown in Fig. 1. As the methodology is based on the 72

exact numerical integration of stiffness coefficients just like the 73

ordinary 3D beam stiffness matrix, increase in computational 74

time and effort is nonexistent. Further, the evaluation of 75

fixed end forces, member forces and beam span deflection 76

remain virtually identical. This, therefore, demonstrated that the 77

stiffness coefficients can be directly and seamlessly integrated 78

to existing structural engineering software without affecting 79

programme architecture and performance. 80

2. Formulations 81

The works of Hailu et al. [11] which used MPC – multi- 82

point constraints – to model the actual geometry and stiffness 83

reductions in RBS systems and, in particular, that of [2], 84

which is the first published attempt on closed-form solution for 85

modelling frames with RBS connections serve as starting points 86

for the current formulations. 87

Please cite this article in press as: Kassegne SK. Development of a closed-form 3-D RBS beam finite element and associated case studies. Engineering Structures(2006), doi:10.1016/j.engstruct.2006.09.010

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ARTICLE IN PRESSS.K. Kassegne / Engineering Structures xx (xxxx) xxx–xxx 3

(a) Top view of beam with RBS connection. (b) Cross section of RBS within theradius-cut region.

Fig. 1. Geometry of beam with radius cut RBS moment connection.

A 12-degrees of freedom beam element with flange1

reductions as shown in Fig. 1 and nodal displacements and joint2

forces shown in Fig. 2 is considered. For simplicity and clarity,3

the particular case of only end ‘1’ deforming while end ‘2’ is4

kept fixed is considered here. This approach will help determine5

the diagonal stiffness sub-matrices [K11] and [K22] indicated in6

Eq. (7.a). The coupling (off-diagonal) sub-matrices [K12] and7

[K21] will be determined through equilibrium equations and8

symmetry considerations, respectively. In standard stiffness-9

based finite element (FE) methodology, there are a number of10

approaches available for deriving the stiffness coefficients like11

the minimization of potential energy, virtual work method, and12

principle of variational calculus. In this case, the virtual work13

method is used where expressions for the virtual works done14

in each of the directions of the six degrees of freedom (per15

node) of a 3D beam element with RBS moment connections16

are derived. Some of the lengthy expressions, which correspond17

to the 2D response such as major axis bending, were already18

reported by Chambers [2]. However, the presence of shear19

deformation in the current formulation introduces significant20

additional terms to those reported by Chambers [2]. These21

additional terms in the FE formulation for 2D responses are,22

therefore, presented here along with the new 3D expressions23

that correspond to torsion and more importantly minor axis24

bending.25

2.1. Virtual work in axial direction26

The internal work done by a unit axial load applied at end27

‘1’ of a beam element with RBS is:28

Wint,axial = 2∫ a

0

f 1x dx

E A+

∫ Le

0

f 1x dx

E A29

+ 2∫ b

−b

f 1x dx

E A[1 − knaxial(√

R2 − x2 + R1)](1.a)30

where naxial =4t fA , E—Young’s modulus, A—area of cross31

section, R—the radius of flange-cut, and t f —thickness of32

flange of beam. Parameters Le, a, b and R1 are defined in the33

Appendix. k = 4 for flange reductions both in the bottom and34

(a) Nodal forces in a 3D beam element.

(b) Degrees of freedom in a 3D beam element.

Fig. 2. RBS element deformations and end forces.

top parts of the beam; otherwise, k = 2 for the case of flange 35

reductions only at the bottom. The joint force, f 1x is shown in 36

Fig. 2 along with the other joint forces. 37

Integrating Eq. (1.a) using Mathematica software [15] 38

or alternatively the trigonometric substitutions adopted by 39

Chambers [2], 40

Wint,axial =f 1x

E A

[2a + Le +

4naxial

(2Ωaxial − γ )

](1.b) 41

where Ωaxial and γ are as defined in the Appendix. 42

On the other hand, the external axial work done by a unit 43

axial load is given as: 44

Wext,axial = (u1− u2). (1.c) 45

Equating the internal and external works: 46

(u1− u2) =

f 1x

E A

[2a + Le +

4naxial

(2Ωaxial − γ )

]. (1.d) 47

Please cite this article in press as: Kassegne SK. Development of a closed-form 3-D RBS beam finite element and associated case studies. Engineering Structures(2006), doi:10.1016/j.engstruct.2006.09.010

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ARTICLE IN PRESS4 S.K. Kassegne / Engineering Structures xx (xxxx) xxx–xxx

For equilibrium, f 1x = − f 2

x . Therefore:1 f 1x

f 2x

=

E A

L R,axial

[1 −1−1 1

]u1

u2

(1.e)2

where L R,axial = 2a + Le +4

naxial(2Ωaxial − γ ). In the limit3

where there is no flange reduction, L R,axial will reduce to L , the4

beam length.5

2.2. Virtual work in torsional direction6

The internal work done by a unit torsional load applied7

at end ‘1’ of a beam element with RBS ignoring torsional8

warping [9,23] is:9

Wint,torsional = 2∫ a

0

m1x dx

G J+

∫ Le

0

m1x dx

G J10

+ 2∫ b

−b

m1x dx

G J [1 − 4n J (√

R2 − x2 + R1)]. (2.a)11

Integration gives:12

Wint,torsional =m1

x

G J

[2a + Le +

4n J

(2ΩJ − γ )

](2.b)13

where n J =4t3

f3J , G is the shear modulus, J is the torsional14

constant, and t f is the top flange thickness. ΩJ and γ are15

as defined in the Appendix. The joint force, m1x represents a16

torsional moment and is shown in Fig. 2 along with the other17

joint forces.18

On the other hand, the external work done by a unit torsional19

load is given as:20

Wext,torsional = (θ1x − θ2

x ). (2.c)21

Equating the internal and external works,22

(θ1x − θ2

x ) =m1

x

G J

[2a + Le +

4n J

(2ΩJ − γ )

]. (2.d)23

For equilibrium, m1x = −m2

x . Therefore,24 m1

xm2

x

=

G J

L R,J

[1 −1−1 1

]θ1

xθ2

x

(2.e)25

where L R,J = 2a + Le +4

n J(2ΩJ − γ ). In the limit, L R,J will26

reduce to L , the beam length.27

2.3. Virtual work in major axis bending28

The internal virtual work done by m1y , moment at end ‘1’29

and the moment due to f 1z , shear at the same end of RBS beam30

is:31

W (int,majorbending)total =

5∑i=1

W iint,majorbending (3.a)32

where i = 1, 5 is the number of distinct geometries in the RBS 33

beam. 34

W 1int,majorbending =

f 1z

E Imaj

∫ a

0−xdx +

m1y

E Imaj

∫ a

0dx (3.b) 35

W 2int,majorbending =

f 1z

E Imaj36

×

∫ b

−b

−(e + x)dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)]37

+m1

y

E Imaj

∫ b

−b

dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)](3.c) 38

W 3int,majorbending =

f 1z

E Imaj

∫ Le

0−(a + 2b + x)dx

+m1

y

E ImajL

∫ Le

0dx (3.d) 39

W 4int,majorbending =

f 1z

E Imaj40

×

∫ b

−b

−( f − x)dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)]41

+m1

y

E Imaj

∫ b

−b

dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)](3.e) 42

W 5int,majorbending =

f 1z

E Imaj

∫ a

0−(L − a + x)dx

+m1

y

E Imaj

∫ a

0dx . (3.f) 43

Adding all the five terms and integrating: 44

(Wint,majorbending)total = −f 1z

E ImajL

(a +

Le

2

)45

+m1

y

E Imaj(2a + Le) 46

−f 1z

E Imaj

L

nmaj,bend

(Ωmaj,bend −

γ

2

)47

+m1

y

E Imajnmaj,bend(2Ωmaj,bend − γ ) 48

(3.g) 49

where Imaj is the major axis moment of inertia, ‘e’ and ‘ f ’ are 50

as shown in Fig. 1. Ωmaj,bend, nmaj,bend and γ are defined in the 51

Appendix. 52

Now, equating external work to internal work done on RBS 53

beam element due to major axis bending and simplifying the 54

ensuing expression: 55

1.θ1y = −

Smaj

E Imajf 1z +

Tmaj

E Imajm1

y (3.h) 56

Please cite this article in press as: Kassegne SK. Development of a closed-form 3-D RBS beam finite element and associated case studies. Engineering Structures(2006), doi:10.1016/j.engstruct.2006.09.010

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ARTICLE IN PRESSS.K. Kassegne / Engineering Structures xx (xxxx) xxx–xxx 5

where1

Smaj = L

(a +

Le

2

)+

L

nmaj,bend

(Ωmaj,bend −

γ

2

)2

Tmaj = 2a + Le +1

nmaj,bend(2Ωmaj,bend − γ ).3

In the limit where there is no flange width reduction, Smaj and4

Tmaj reduce to L2/2 and L respectively.5

2.4. Virtual work done in major axis shear including shear6

deformation7

The virtual work done due to f 1z , a unit shear force applied8

in the major axis direction at end ‘1’ of a beam with RBS9

connection is:10

W (int,majorshear)total =

5∑i=1

W iint,majorshear (4.a)11

where i = 1, 5 is the number of distinct geometries in the RBS12

beam.13

W 1int,majorshear =

f 1z

E Imaj

∫ a

0x2dx −

m1y

E ImajL

∫ a

0xdx

+f 1z

G Asmaj

∫ a

0dx (4.b)14

W 2int,majorshear =

f 1z

E Imaj

∫ b

−b

(e + x)2dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)]15

−m1

y

E Imaj

∫ b

−b

(e + x)dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)]16

+f 1z

G Asmaj

∫ b

−bdx . (4.c)17

Note that the shear deformation terms in the flange reduction18

area see no reductions because the major shear area is19

essentially the area of the web which does not get reduced in20

RBS connections.21

W 3int,majorshear =

f 1z

E Imaj

∫ Le

0(a + 2b + x)2dx22

−m1

y

E Imaj

∫ Le

0(a + 2b + x)dx +

f 1z

G Asmaj

∫ Le

0dx (4.d)23

W 4int,majorshear =

f 1z

E Imaj

∫ b

−b

( f +x)2dx

[1−4nmaj,bend(√

R2−x2+R′)]24

−m1

y

E Imaj

∫ b

−b

( f + x)dx

[1 − 4nmaj,bend(√

R2 − x2 + R′)]25

+f 1z

G Asmaj

∫ b

−bdx (4.e)26

W 5int,majorshear =

f 1z

E Imaj

∫ a

0(L − a + x)2dx27

−m1

y

E Imaj

∫ a

0(L − a + x)dx +

f 1z

G Asmaj

∫ a

0dx . (4.f)28

Now, adding all the five terms and integrating, 29

(Wint,majorshear)total =Umaj

E Imajf 1z +

Hmaj

G Asmajf 1z −

Smaj

E Imajm1

y 30

(4.g) 31

where 32

Umaj =

[2a3

3+

Le3

3+ aLe(Le + a) + 2bLe(L − 2b) 33

+ aL(L − a)

]+

14nmaj,bend

−2γ (2R2

+ e2+ f 2) 34

+ 4Ωmaj,bend

[e2

+ f 2+ 2R2

(1 −

1

α2

)]+ 2λmaj,bend

35

Hmaj = L 36

Smaj = L

(a +

Le

2

)+

L

nmaj,bend

(Ωmaj,bend −

γ

2

). 37

λ is defined in the Appendix. In the limit, Umaj and Smaj will 38

reduce to L3/3 and L2/2 respectively. 39

Now, equating external work to internal work done on RBS 40

beam element due to major axis shear: 41

1.w1=

(Umaj

E Imaj+

Hmaj

G Asmaj

)f 1z −

Smaj

E Imajm1

y . (4.h) 42

In matrix form, the equations of equilibrium, i.e. Eqs. (3.h) 43

and (4.h) could be re-written together as: 44(

Umaj

E Imaj+

Hmaj

G Asmaj

)−

Smaj

E Imaj

−Smaj

E Imaj

Tmaj

E Imaj

f 1z

m1y

=

w1

θ1y

. (4.i) 45

Inversion of the above matrix yields, 46f 1z

m1y

=

E Imaj

(Umaj + Φmaj)Tmaj − SmajSmaj47

×

[Tmaj SmajSmaj (Umaj + Φmaj)

]w1

θ1y

(4.j) 48

where Φmaj, the major axis shear deformation term is given as: 49

Φmaj =Hmaj E Imaj

G Asmaj. 50

2.5. Virtual work in minor axis bending 51

The virtual work done in the minor axis is very much 52

analogous to that of the major axis with the exception that the 53

net minor axis moment of inertia (Imin) in the flange reduction 54

area is calculated differently with the reductions assuming the 55

third power of the removed flange width (see the Appendix and 56

Fig. 1(b)). Now, following the same procedure as that of the 57

major axis bending, the final expression for the work done in 58

minor axis bending is of the form: 59

(Wint,minorbending)total = −f 1y

E IminL

(a +

Le

2

)60

Please cite this article in press as: Kassegne SK. Development of a closed-form 3-D RBS beam finite element and associated case studies. Engineering Structures(2006), doi:10.1016/j.engstruct.2006.09.010

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ARTICLE IN PRESS6 S.K. Kassegne / Engineering Structures xx (xxxx) xxx–xxx

+m1

z

E Imin(2a + Le)1

−f 1y

E Imin

∫ b

−b

(e + x) + ( f − x)dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)22

+m1

z

E Imin

∫ b

−b

2dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)2 (5.a)3

where bx =√

R2 − x2 + R1 (as given in the Appendix).4

Now, equating external work to internal work done on RBS5

beam element due to minor axis bending and simplifying the6

ensuing expression gives:7

1.θ1z = −

Smin

E Iminf 1y +

Tmin

E Iminm1

z (5.b)8

where Smin and Tmin can be evaluated through numerical9

quadrature as:10

Smin = L

(a +

Le

2

)+

∫ b

−b

(e + f )dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)211

Tmin = 2a + Le +

∫ b

−b

2dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)2 .12

In the limit where there is no flange width reduction, Smin13

and Tmin will reduce to L2/2 and L respectively.14

2.6. Virtual work in minor axis shear including shear15

deformation16

The same procedures used for major axis shear case are17

repeated here for deriving the virtual work done, the end18

moments and end shear forces. Similar expressions for the19

minor axis bending and shear could be written by replacing the20

major axis properties with minor axis properties. However, it21

has to be noted that the minor shear area is not neglected in the22

minor axis direction unlike the major axis case and, therefore,23

additional shear terms will show up in the virtual work as shown24

below. For brevity, only the final expressions are given here25

taking into account the additional terms due to reduction in26

minor shear area.27

(Wint,minorshear)total =f 1y

E Imin

[2a3

3+

Le3

328

+ aLe(Le + a) + 2bLe(L − 2b) + a(L − a)

]29

+f 1y

G As,min[2a + Le] −

m1z

E Imin

(a +

Le

2

)30

+f 1y

E Imin

∫ b

−b

(e + x)2+ ( f + x)2dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)231

+f 1y

G As,min

∫ b

−b

2dx

(1 − 4nmin,shearbx )32

−m1

z

E Imin

∫ b

−b

(e + x) + ( f + x)dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)2 . (6.a) 33

Simplifying, 34

(Wint,minorshear)total =Umin

E Iminf 1y +

Hmin

G Asminf 1y −

Smin

E Iminm1

z 35

(6.b) 36

where Smin is defined in Section 5 and Hmin is as given below: 37

Hmin = 2a + Le +4

nmin,shear(2Ωmin,shear − γ ) 38

Umin can be evaluated through numerical quadrature as: 39

Umin =

[2a3

3+

Le3

3+ aLe(Le + a) 40

+ 2bLe(L − 2b) + a(L − a)

]41

+

∫ b

−b

(e + x)2+ ( f + x)2dx

1 −4t fImin

b3

x12 + bx

(b f −bx

2

)2 42

Ωmin,shear, nmin,shear and γ are defined in the Appendix. In the 43

limit, Umin and Hmin will reduce to L3/3 and L , respectively. 44

Following the same procedure as in the major axis bending 45

and shear, the stiffness matrix that relates the minor axis shear 46

and moment at end ‘1’ with the corresponding transverse 47

displacement and rotation at the same end is derived. 48f 1y

m1z

=

E Imin

(Umin + Φmin)Tmin − SminSmin49

×

[Tmin SminSmin (Umin + Φmin)

]v1

θ1z

(6.c) 50

where Φmin, the minor axis shear deformation term is given as: 51

Φmin =Hmin E Imin

G Asmin. 52

The displacement-joint force relationships for the release 53

of joint ‘2’ are identical to the relationships for the release 54

of joint ‘1’. These are represented as [K22] in Eq. (7.a). The 55

only relationships required to complete the evaluation of the 56

whole 12×12 stiffness matrix is, then, the sub-matrix [K12] that 57

relates displacements and joint forces at joints ‘1’ and ‘2’ when 58

both joints are released. This sub-matrix can be derived through 59

equilibrium equations such as those given by Eq. (7.b). 60[[K11] [K12]

Symm. [K22]

]∆1

∆2

=

F1

F2

(7.a) 61

m2y = f 1

z L − m1y (7.b) 62

where ∆1 and ∆2

are each arrays of six displacement 63

components at joints ‘1’ and ‘2’ and F1 and F2

are arrays of 64

the six joint force components at joints ‘1’ and ‘2’, respectively. 65

The final 12×12 stiffness matrix and stiffness equation 66

based on Tiomoshenko’s Beam Theory – as shown in Box I – 67

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K11 −K11K22 K26 −K22 K26

K33 K35 −K33 K35K44 −K44

K55 −K35 K5,11K66 −K26 K6,12

K11Sym. K 22 −K26

K33 −K35K44

K55K66

u1

v1

w1

θ1x

θ1y

θ1z

u2

v2

w2

θ2x

θ2y

θ2z

=

f 1x

f 1y

f 1z

m1x

m1y

m1z

f 2x

f 2y

f 1z

m1x

m1y

m1z

Box I.

for a beam element with RBS connections at both ends is, then,1

given by combining Eqs. (1.e), (2.e), (4.j), (6.c) and (7.b): see2

Box I where3

K11 =E A

L R,axial,4

K22 =Tmin E Imin

(Umin + Φmin)Tmin − SminSmin,5

K26 =Smin E Imin

(Umin + Φmin)Tmin − SminSmin6

K33 =Tmaj E Imaj

(Umaj + Φmaj)Tmaj − SmajSmaj,7

K35 =Smaj E Imaj

(Umaj + Φmaj)Tmaj − SmajSmaj8

K44 =G J

L R,Z9

K55 =(Umaj + Φmaj)E Imaj

(Umaj + Φmaj)Tmaj − SmajSmaj,10

K5,11 =(SmajL − (Umaj + Φmaj))E Imaj

(Umaj + Φmaj)Tmaj − SmajSmaj11

K66 =(Umin + Φmin)E Imin

(Umin + Φmin)Tmin − SminSmin,12

K6,12 =(SminL − (Umin + Φmin))E Imin

(Umin + Φmin)Tmin − SminSmin.13

3. Verification of 3D RBS finite element and example14

problems15

The accuracy of the developed algorithm and program is16

tested using the following examples: cantilever and fixed beams17

with RBS connections, 2D frames with shear deformation18

under lateral loads, 3D frame with eccentric lateral loads,19

gravity loads and torsionally irregular framing, and 3D frames20

eigenvalue analysis. The quantities investigated are: effect on21

story and displacements, drift and mode shapes. Further, the22

increases in rotation stiffness coefficients (see Box I) in major23

and weak axis bending for some commonly used wide-angle24

beam sections are investigated.25

The width of the radius cut, ‘c’ as shown in Fig. 1 is gives 26

as: 27

c = 0.5 ∗ (1 − r) ∗ BeamWidth (8.a) 28

where ‘r ’ varies from 0.5 to 0.9 (which correspond to 50% to 29

10% cut). In general, the percentage of flange removal is taken 30

as (2c/beamwidth) ∗ 100. 31

On the other hand, there are two sets of recommendations 32

for the dimensions of the parameters ‘a’, and ‘b’ (Fig. 1). The 33

recommendations of Moore et al. [16] and SAC [18] specify the 34

following values: 35

a = 50%–75% beam flange width; (8.b) 36

b = 65%–85% beam depth. (8.c) 37

Iwankiw and Carter [14] recommend the following values. 38

a = 25%–50% of the beam depth; (8.d) 39

b = 75%–100% of the beam depth. (8.e) 40

In this study, information on which recommendation is based 41

is provided for each example solved here. For simplicity, in all 42

the examples reported in this study, the percentage reductions 43

in RBS width are the same for all the beams in a given structure. 44

The values of ‘a’ and ‘b’, of course, depend on the beam flange 45

width and depth which are a function of the size of the ‘WF’ 46

beam selected. In all the examples reported here, structural 47

steel is assumed to have E (Young’s Modulus) of 29 000 ksi 48

(2.0 × 1011 Pa) and Poisson’s ratio of 0.29. 49

3.1. Cantilever and fixed RBS element 50

The accuracy of the element developed in this study is 51

verified through a comparison with results from [1,8]. A single 52

cantilever beam of 20 f (6.096 m) length with a 50% radius cut 53

RBS connections at both ends was considered. The dimensions 54

of ‘a’ and ‘b’ were determined using the recommendation of 55

[16] where a = 9.675 in. (245.75 mm) and b = 21.029 in. 56

(534.14 mm). The size of the beam is W24 × 146 with 57

Imaj = 4569.57 in.4 (1.902 × 109 mm4), A = 43 in.2 58

(27 742 mm2), and Asmaj = 16.081 in.2 (10 374.8 mm2). The 59

beam is subjected to a line of point loads at the free end as 60

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(a) Contour of deflected shape without RBS. (b) Contour of deflected shape with RBS.

Fig. 3. Finite element modeling (ANSYS) of a cantilever beam with RBS connections under major axis bending due to a line load at the tip.

Table 1Comparison of results from present element with FEA programmes

No cut-out 50% radius-cutFEA Present FEA Present

Cantilever beam under major-axis bending(Fig. 3)

Tip disp. 1.0834 in (27.518 mm) 1.0833 in (27.516 mm) 1.1646 in (29.581 mm) 1.1645 in(29.578 mm)

% increase – – 7.5% 7.55%

Cantilever beam under minor-axis bending(Fig. 4)

Tip disp. 1.2236 in (31.080 mm) 1.2234 in (31.074 mm) 2.9455 in (74.816 mm) 2.9452 in(74.808 mm)

% increase – – 140.72% 140.75%

Fixed beam under minor-axis bending(Fig. 5)

Cent. disp. 0.8188 in (20.797 mm) 0.8186 in (20.792 mm) 1.1189 in (28.420 mm) 1.1186 in(28.412 mm)

% increase – – 36.65% 36.64%

(a) Contour of deflected shapewithout RBS.

(b) Contour of deflected shape with RBS.

Fig. 4. Finite element modeling of a cantilever beam with RBS connections under weak axis bending due to a line load at the tip.

shown in Fig. 3. The total value of the point loads is −30 kips1

(133.4 kN) acting downward. The ANSYS model consists of2

10-node tetrahedral structural solid (SOLID92 element) with3

about 11 500 elements for both models with and without RBS4

moment connections. The comparisons are given in Table 15

where the tip vertical displacements at the free end predicted6

by the RBS element developed in this study compare closely7

with those predicted by ANSYS.8

The same cantilever was then subjected to a weak axis9

bending under a line of tip loads totaling 3 kips (Fig. 4).10

The mesh for the model with no RBS consisted of 18 195 11

tetrahedral structural solid elements whereas the mesh for the 12

model with RBS connections had 18 416 elements. Table 1 13

summarizes the tip displacements obtained from a finite 14

element analysis program [8] and the element developed in 15

this study with both the new developed 3D element and 16

FEA package giving increase in out-of-plane tip displacement 17

in excess of 140%. This comparison demonstrates that the 18

increases in weak axis displacements in the presence of RBS 19

connections are considerable and much higher than those 20

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(a) Contour of deflected shape without RBS. (b) Contour of deflected shape with RBS.

Fig. 5. Finite element totaling of a fixed beam with RBS connections under weak axis bending due to a uniform line load.

Fig. 6. A 2-D frame with 2 bay and 6 stories under lateral loads. Equal bay-width and story heights.

corresponding to major axis displacements. Further, the same1

beam was reanalyzed with fixed end boundary conditions for2

both ends of the beam (Fig. 5). A lateral (weak axis) line3

load of 1 kip/ft (14.6 kN/m) was applied on both models with4

and without RBS connections. As summarized in Table 1, the5

FEA analysis using FEMLAB gives a close agreement with the6

present element in predicting the increase in weak axis bending;7

both showing about 36% increase in the center-span weak axis8

displacement due to the presence of RBS connections. These9

close comparisons in both major axis as well as minor axis10

bending between the new element and the commercial FEA11

packages [1,8] demonstrate the accuracy of the 3D RBS beam12

element developed here.13

3.2. Drift in 6-story 2-bay 2D frame14

This frame shown in Fig. 6 is very similar to the frame15

analyzed by Chambers [2] who reported a drift increase of16

10.6% with a 40% flange-area reduction. Both column and17

beam sections are shown in the figure. The beam sizes vary 18

from W24×68 at the roof to W30×235 in the bottom three 19

stories whereas the top two floors have columns of size 20

W24×279 and the bottom four floors have columns of size 21

W24×370. The model solved by Chambers [2] is identical to 22

this model except that the bottom four floors have columns of 23

size W24×492 which is rarely used. The lateral loads are 85.71, 24

71.43, 57.14, 42.86, 28.57, 14.29 kips (1 kip = 4.448 kN) 25

applied at the left ends of the frame starting from top floor 26

and going down. The beams are free to deflect in the axial 27

direction, as there is no diaphragm. A maximum reduction in 28

flange width of 50% and b = 0.5d, and a = 0.25d are used, 29

as per Iwankiw and Carter [14] recommendations. Note that 30

while the flange reductions in this example were varied from 31

10%–50% to demonstrate the increase in lateral displacements 32

with different flange reductions, recommendations such as [18] 33

limit the minimum flange reductions to 40%. The results are 34

given in Table 2 which shows a maximum increase of 11.04% 35

in lateral roof displacement using the current formulations. The 36

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Table 2Verification of present algorithm for modeling effect of RBS on a two-dimensional frame

No cut-out 10% cut-out 30% cut-out 50% cut-out

Roof displacement 2.1583 in (54.82 mm) 2.1949 in (55.752 mm) 2.3411 in (59.463 mm) 2.3829 in (60.5279 mm)% increase – 1.7% 8.47% 11.04%

(a) Effect of ratio of beam span to depth ratio on K33 and K55stiffness coefficients. r = 0.5 (50%).

(b) Effect of flange area reduction ratio on K33 and K55stiffness coefficients. r = 0.6 (40% reduction).

Fig. 7. Decrease in stiffness coefficients for major axis bending in W27×336 beam for a variety of span to depth and flange reduction ratios. (a) Represents [14]and (b) Represents [16] recommendations.

small discrepancy between the two studies are due to shear1

deformation in the current formulation reported here and –2

importantly – the use of different column sizes at the bottom3

four floors as discussed above.4

3.3. Effect of RBS connections on beam stiffness coefficients5

The quantitative assessment of the various stiffness6

coefficients for a beam element and their reduction in the7

presence of RBS connections is of practical interest. For a8

beam of W27×336 section with Imaj = 14 500 in.4 (6.035 ×9

109 mm4), Imin = 1170 in.4 (4.87 × 108 mm4), A = 98.7 in.210

(63 677 mm2), Asmaj = 37.8 in.2 (24 387 mm2), and Asmin =11

55.27 in.2 (35 658 mm2), for example, a detailed analysis is12

carried out to determine the magnitude of stiffness coefficients13

reductions in major axis and minor axis bending in the presence14

of RBS connections of various flange area cut ratio and beam15

length to depth ratios. The stiffness coefficients of practical16

interest are K33 and K55 given in Box I which represent the17

joint shear and rotational stiffnesses respectively for major axis18

bending. For minor axis bending, K22 and K66 given in Box I,19

which represent the joint minor shear and minor rotational20

stiffnesses respectively, are of interest.21

Fig. 7 shows the percentage decrease of the major axis22

rotational and shear stiffness coefficients in the presence of23

RBS connections at both sides as compared to the full beam24

section stiffness coefficients. For L/d of 10.0 and a =25

7.5 in. (190.5 mm) and b = 15.0 (381.00 mm) which were26

determined using recommendation of Iwankiw and Carter [14],27

a maximum of about 15.5% reduction is observed for the28

shear stiffness coefficient (K33), and 12.5% reduction for the29

rotational stiffness coefficient (K55). As the length of the beam 30

increases, these reductions in stiffnesses typically exhibit a 31

decreasing trend. However, at small L/d ratios (less than 10), 32

the contribution from shear deformation becomes significant 33

and it slows the overall reduction in stiffnesses due to the fact 34

that there is no reduction in shear area in major axis bending 35

as explained in the formulations section. For L/d ratios below 36

8 or so – ratios quite uncommon in realistic frames – the shear 37

deformations are so dominant that the stiffness reductions do 38

actually start decreasing for decreased L/d ratios. Fig. 7 shows 39

this rather unexpected trend for low L/d ratios. 40

For the same flange area reduction parameters, the minor 41

axis bending results show a maximum of 67% reduction 42

in shear stiffness coefficient (K22) and 65% reduction for 43

the rotational stiffnesses (K66) as shown in Fig. 8. In the 44

case of beams with RBS flange reductions only at the 45

bottom flanges (as in retrofitting), these reductions will go 46

down by half to about 33% and 32% respectively, assuming 47

a negligible warping effect. These rather large minor axis 48

stiffness coefficients reductions are due to the fact that the 49

decrease in Iyy (minor axis moment of inertia) in RBS beams 50

is not linear (as in the case of major axis bending); but cubic in 51

the presence of flange cuts as shown in the formulation section 52

(see also the Appendix). A comparison of the reduction on the 53

integrated minor and major axis inertias for a variety of wide- 54

angle beam sizes is given in Fig. 11 which demonstrates that for 55

W24×68 section, for example, the minor axis moment of inertia 56

(integrated over the length of the radius-cut region) reduces by 57

about 23% whereas the major axis moment of inertia shows a 58

decrease of only 0.5% for r = 0.5. For deeper sections such as 59

W36×150, the reductions in integrated moments of inertia are 60

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(a) Effect of ratio of beam span to depth ratio on K22 and K44stiffness coefficients. r = 0.5 (50%).

(b) Effect of flange area reduction ratio on and K44 stiffnesscoefficients stiffness reductions.

Fig. 8. Decrease in stiffness coefficients for minor axis bending in W27×336 beam for a variety of span to depth and flange reduction ratios. (a) Represents [14]and (b) Represents [16] recommendations.

Fig. 9. Decrease in stiffness coefficients for minor axis bending in W27×336beam with different RBS cut length to beam span ratios. In case 1, a = 7.5 in.

(190.5 mm) and b = 15.0 (381.00 mm) with r = 50%; whereas in case 2, anda = 15.0 in. (381.00 mm) and b = 11.25 in. (285.75 mm) and 30% cut.

down to 9% for minor axis and 0.4% for major axis for r =1

0.5. For steel buildings with a composite floor deck, the floor2

provides a lateral support and a composite action to the beams3

whereby the actual weak axis bending may not be as much4

as predicted by this formulation. However, it has to be noted5

that the bottom flanges of these beams are still unsupported6

by the floor; making reduction in weak axis bending stiffness7

still an issue of concern. Further, for beams located at roofs8

which are typically made of light metal deck and beams in9

areas of slab opening where there is no substantial diaphragm10

support, the large reduction in weak axis bending stiffness could11

potentially be a design concern. Some of the adverse effects12

of the reduction in weak axis bending stiffness include high13

torsional moment requirement on supporting columns and also14

potential lateral buckling of such beams which is a function of15

the weak-axis stiffness of flanges in compression regions [6].16

Figs. 7(b) and 8(b) show the effect of ratio of flange17

area reduction, r , on the stiffness coefficients where an18

Fig. 10. Decrease in stiffness coefficients for torsional and axial stiffness inW24×76 beam with different RBS cut length to beam length ratios. In case 1,a = 7.5 in. (190.5 mm) and b = 15.0 (381.00 mm) with r = 50%, whereas incase 2, and a = 15.0 in. (381.00 mm) and b = 11.25 in. (285.75 mm) and 30%cut.

increase in r (hence less cut) gives a nonlinearly decreasing 19

reduction in stiffnesses. Figs. 7 and 8 further compare the 20

stiffness reductions obtained for the two commonly used 21

recommendations of [14,16] where it is shown that Moore 22

et al. [16] give a slightly conservative results. Fig. 9 shows 23

the effect of selection of the flange area reduction parameters. 24

A 30% flange area cut and a narrower RBS region positioned 25

further from the adjoining column at 15 in (381 mm) results 26

in minor axis stiffness coefficient reduction of 45%, which 27

is about 20% down from the case with a wider RBS region 28

positioned nearer to columns. The effect of the presence of RBS 29

moment connections on axial and torsional stiffness coefficients 30

(K11) and (K44) is shown in Fig. 10 where the reductions are 31

shown to be below 10% even for a 50% flange area reductions. 32

In Fig. 11, the effect of compactness of a rolled beam section 33

on the reduction on the integrated minor axis moment of inertia, 34

Imin, over the whole RBS region is demonstrated. The most 35

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Fig. 11. Decreases in major and minor moments of inertia for three beam sizeswith different compactness and flange area reduction ratios (50%, 40%, 30%,20%, etc.).

reduction in Imin is observed in W24×68 sections (23%, with1

Imin = 70.4 in.4 or 2.93 × 107mm4). A W30×235 section2

(Imin = 855 in.4 or 3.559 × 108 mm4) experiences only a3

maximum of 3% reduction in Imin even for a 50% flange-area4

reduction.5

3.4. Drift in 3-dimensional frame system with shallow columns6

This is a 6-story building with non-symmetric frame layout7

as shown in Fig. 12. All the bays are 25 f (7.62 m) wide with8

a uniform story height of 13 f (3.962 m). For all frames, the9

columns sizes are W14×426 for all the top five stories and10

W14×455 for the ground floor. These sizes are pre-qualified11

as per the FEAM recommendations [7]. For frames 1–4, the12

beam sizes are W24×68 at the top story and W24×192s for13

all other stories. For frame 5, the beam sizes are W24×192 for14

all stories. Two lateral load cases are considered with 125, 100,15

80, 60, 40, and 20 kips (1 kip = 4.448 kN) applied in both16

directions at location of 100 ft, 50 ft (30.50 m, 15.25 m). A17

uniform line load of gravity loads of 3 kips/ft (43.75 kN/m)18

is applied at all the beams in Frames 2 and 3. Radius cut19

of 50% based on Iwankiw and Carter recommendations [14]20

are used. In this building, the beams at each story level are21

constrained to move with the diaphragms; therefore there are22

no axial deformations and minor axis bending in the beams. In23

addition, full rigid end zones are considered.24

The results are summarized in Fig. 13 which shows increases25

in drifts at each of the nodes in the building model in the26

presence of RBS for both lateral loads only and a combination27

of lateral and gravity loads in x- and y-directions. There are28

a total of 84 nodes with 14 nodes at each story as shown in29

Fig. 12. The nodes are numbered from top to bottom; nodes30

from 1 to 14 representing nodes in the roof. The figure shows31

that drifts at the nodes increase by amounts varying from a32

maximum of about 13% at the roof level to 7.5% at the second33

and first floors for a load combination of gravity and lateral34

loads (1.2DL + 1.4E). For x-direction lateral loads alone, the35

maximum drift increases are about the same at 13% at the roof36

Fig. 12. Plan view of unsymmetrical 3D frame with shallow columns undergravity and earthquake lateral loads. Node numbers are circled.

Fig. 13. Increase in drift in an unsymmetrical 3D frame with shallow columnssubjected to lateral and gravity loads and with RBS moment connection.

level and within 6% at the first floor. This shows that for these 37

particular levels of gravity loads and frame configurations, the 38

effect of combining gravity and lateral loads are not substantial 39

even though each of the load cases produced diaphragm 40

rotations in different directions. The effects of different radius 41

cut ratios and bay widths is demonstrated in Fig. 14 which 42

shows that the drift increases could go higher to 13.5% if the 43

bay width is reduced from 25 f (7.62 m) to 20 f (6.096 m) for 44

a 50% flange area reduction. For lower flange area reductions, 45

the drift increases go down to 10% and less, as expected. 46

3.5. Drift in 3-dimensional unsymmetrical frame system with 47

deep columns 48

This is a 6-story building frame with symmetric lateral 49

frames in the x- and nonsymmetrical frames in the y-directions 50

as shown in Fig. 15. Deep columns (W24s) are used in this 51

example to assess the effect of RBS connections on such 52

framing systems. The objective of this example is primarily 53

to investigate if the increases in story drifts in frames with 54

deep columns follow the same trends as in shallower columns. 55

Incidentally, recent research argues that deep columns – which 56

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Fig. 14. Increase in drift at each story for various radius-cut ratios (30, 40, and50%) and different bay widths (L1 and L2) for unsymmetrical 3D frame withshallow columns subjected to lateral and gravity loads and with RBS momentconnection. L1 = 25 f (7.62 m), L2 = 22 f (6.7 m).

Fig. 15. Plan view of unsymmetrical 3D frame with deep columns undergravity and earthquake lateral loads. Node numbers are circled.

are not pre-qualified by present recommendations – behave very1

well inelastically [20,17] contrary to earlier research that had2

suggested instability failures [22]. The debate is expected to3

continue as results obtained from such research as this one4

already indicate substantial reductions in weak axis bending5

of beams with RBS connections which will put additional6

torsional requirements on the framing columns. In light of this,7

therefore, it is believed that a review of the elastic behaviour8

of such systems, particularly from drift perspective, has some9

merit.10

In the x-direction, two 2-bay parallel frames (1 and 2) –11

with equal bay spacing of 24 f (7.315 m) – are used at the12

outer perimeter edge. In the y-direction, 2-bay frames (3 and13

4) are used at the outer edge of the building perimeter with14

the same 24 f spacing. The building dimensions are 120 ft x15

96 ft (36.58 m × 29.26 m) with a uniform story height of16

13 ft (3.96 m). The elevation of Frames 1, 2 and 3 is given17

in Fig. 16 where beam sizes vary from W30×235 (bottom18

floor) to W24×62 (roof). All the columns in all the frames19

are W24×335. Frame 4 has identical column sizes as Frame 20

1, but all the beams are W24×68s; thus introducing stiffness 21

irregularity. 22

In this SMRF, independent lateral loads in the x- and y- 23

directions (with total base shear of 420 kips respectively) were 24

applied. The x-direction lateral loads are (120, 100, 80, 60, 40, 25

and 20 kips) (1 kip = 4.448 kN) applied at a location of 60 ft, 26

8 ft (28.29 m, 2.44 m). The same load pattern is applied in the 27

y-direction as well at a location of 100 ft, 0 ft (30.5 m, 0 m). A 28

uniform line load of gravity loads of 3 kips/ft (43.75 kN/m) 29

is applied at all the beams in Frames 2 and 3. Further, an 30

eigenvalue analysis is also carried out to determine the effect 31

of RBS moment connections on the dynamic characteristics 32

of the frame. Radius cuts of 10%, 40% and 50% were used. 33

The radius-cut parameters are based on Iwankiw and Carter 34

recommendations [14]. The beams at each story level are 35

constrained to move with the diaphragms; therefore there are 36

no axial deformations and minor axis bending in the beams. 37

Full rigid end zones are considered. 38

The results are summarized in Figs. 17–19. There are a total 39

of 66 nodes with 11 nodes at each story as shown in Fig. 13. 40

The nodes are numbered from top to bottom; nodes from 1–11 41

representing nodes in the roof. For a load combination of dead 42

load and x-direction lateral load case, Fig. 17 shows that a 43

maximum of about 13.5% increase in drifts at the nodal points 44

is observed for RBS connections with radius cut of 50%. The 45

lateral loads by themselves result in about 13% drift increase 46

at the roof level which drops to about 7% in the first floor. As 47

shown in Fig. 19, in the presence of RBS moment connections, 48

the y-direction loads cause even higher drift increases within 49

the range of 15% at the roof level for a combination of gravity 50

and lateral loads (1.2DL + 1.4E) and about 14.4% for a 51

lateral load case. The drift increases are observed to decrease 52

at the lower stories reaching drift values as low as 7%. The 53

effect of beam sizes on the drift increases – while keeping 54

column sizes unaltered – is shown in Fig. 18 where the deeper 55

and stiffer W30×235 beams (Imaj = 11 700 in.4 or 4.87 × 56

109 mm4) are replaced by the less-stiff W24×250 wide-flange 57

sections (Imaj = 8490 in.4 or 3.53 × 109 mm4). Dropping the 58

beam sizes to W24×250, increases the drift due to x-direction 59

earthquake load in the presence of RBS moment connections 60

from 13% to about 15%. This increase is due to the fact that 61

the contribution of beams towards the lateral displacements 62

and drifts increases when smaller sizes are used. This trend of 63

increased degradation of lateral stiffness in frames with large 64

columns and smaller beam sizes with RBS moment connection 65

is an important demonstration of the inadequacy of the ‘one- 66

size-fits-all’ approach of 9% drift increase recommended by 67

FEMA [7] and SAC [18]. 68

The eigenvalue comparisons indicate a modest average 69

increase of 6% for a 50% radius-cut. This smaller increase 70

in periods is as expected because the period of a structure is 71

directly proportional to the square root of the stiffness, which 72

suggests that the increase will be increasing not linearly but at 73

the slower rate of the square root of the decrease in stiffness. 74

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Fig. 16. Elevation view of Frames 1, 2 and 3. Equal bay-widths and story heights.

Fig. 17. Effect of RBS connection on story displacements for lateral loads inx-direction and a combination of gravity and lateral loads.

Fig. 18. Effect of beam size on drift increases in x-direction for SMRFwith RBS connections. Column sizes are kept constant and 1.2DL + 1.4Ecombination is used.

Fig. 19. Effect of RBS connection on story displacements for lateral loads iny-direction and a combination of gravity and lateral loads.

4. Conclusions 1

The study shows that the presence of RBS moment connec- 2

tions in SMRF could cause increase of story displacements as 3

much as 15% in cases where diaphragm rotation is present un- 4

der gravity and lateral loads. Approximate analysis tools based 5

on 2- dimensional frames where there were no rotations had re- 6

ported drifts of about 10%. The additional drifts reported in this 7

study are, therefore, caused by consideration of shear deforma- 8

tions and diaphragm rotations—quantities not studied in prior 9

research work. The effect of combining gravity and lateral loads 10

– at least for cases reported here – are not substantial in terms 11

of increasing story drifts. Results from this study also indicate 12

that frames with both shallow and deep columns are sensitive 13

to drift increase due to the introduction of RBS connections. 14

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Further, this study also highlights that the assumption that1

a 10% increase in story displacements translates to an equal2

amount of drift is erroneous since the effect of RBS connections3

at each of the story levels is different resulting in a potentially4

higher drift increases.5

The study also predicts that the decrease in weak axis6

bending stiffness of beams with RBS connections on both sides7

– and unsupported laterally – could be as high as 67%. For8

beams subjected to weak axis bending where either the bottom9

flange or both flanges of these beams is still unsupported by the10

floor and for beams located at roofs which are typically made11

of light metal deck, this drastic decrease in weak axis bending12

stiffness could have adverse effects due to the large reduction13

in stiffness available to resist this bending. Some of these14

adverse effects include high torsional moment requirement on15

supporting columns and also potential lateral buckling of such16

beams in the laterally unsupported regions. Work in extending17

the current formulations to assessing elastic lateral torsional18

buckling (LTB) in beams with RBS moment connections is19

already under progress.20

This study that takes into account shear deformation and21

effect of diaphragm rotations and gravity and lateral loads22

combination is considered more realistic and based on rational23

analysis and, through numerous examples reported here and24

compared with commercial FEA programmes, has been verified25

to give accurate results.26

Therefore, it is recommended that the analysis and design of27

special moment resisting frames (SMRF) with RBS moment28

connections using the traditional 9% and 10% increases be29

reviewed in light of these results. With this new work,30

rational analysis based on realistic modelling of the member31

stiffness based on its reduced section is now available to32

structural engineers for modelling 3-dimensional frames with33

Timoshenko’s Beam Theory and its use should be encouraged34

for routine structural design practice.35

Uncited references36

[3], [12], [13] and [21].37

Acknowledgements38

The contribution of Young Li in using a mathematical39

software for symbolic manipulation of some of the integrals40

and Jeff Milton in independently verifying some of the results41

is gratefully acknowledged.42

Appendix43

44

R =

14

[(1 − r)2b2

f + b2]

(1 − r)b f45

R1= (1 − r)

b f

2− R46

Ωsub =

tan−1[√

1+αsub1−αsub

tan γ2

]√

1 − α2sub

47

γ = sin−1(b/R) 48

αsub =4Rnsub

1 − 4nsub R1 49

λ majmin

= R2

[γ +

sin 2γ

2+

2α maj

min

(sin γ +

γ

α majmin

)]50

nmaj,bend =1

Imaj

t3

f

12+ t f

((d − t f )

2

)2

51

nmin,shear =4t f

Asmin

(56

)52

nmin,bend =1

Imin

t f

b3x

12+ t f bx

(b f − bx

2

)2

, 53

where bx =

√R2 − x2 + R1. 54

The subscript ‘sub’ indicates that the expression is generic 55

and is valid by replacing it with the appropriate term. For 56

example, for axial stiffness, the subscript ‘sub’ is replaced by 57

‘axial’; for torsional stiffness, major axis bending stiffness, 58

minor axis bending stiffness, major axis shear stiffness, minor 59

axis shear stiffness, it is replaced by ‘J ’, ‘major,bend’, 60

‘minor,bend’, ‘major,shear’, and ‘minor,shear’. 61

References 62

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stiffness of moment frames. Journal of Structural Engineering ASCE 65

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[18] SAC Joint venture. Recommended seismic design criteria for new steel8

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2000.12

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Please cite this article in press as: Kassegne SK. Development of a closed-form 3-D RBS beam finite element and associated case studies. Engineering Structures(2006), doi:10.1016/j.engstruct.2006.09.010